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Let $X_1, X_2,\cdots$ be a sequence of independent, identically distributed random variables and $\displaystyle S_n=\sum_{i=1}^{n}X_i$. Then $EX_{1} <0$ if, and only if , $\displaystyle\lim_{n \rightarrow \infty } S_{n} = -\infty \quad\text{a.s.}$

If $EX_{1} < 0$ then $\displaystyle\lim_{n \rightarrow \infty } S_{n} = -\infty \quad\text{a.s.}$ using the strong law of large numbers. This shows one of the implications.

Now, suppose that $EX_{1} \geq 0$. If $EX_{1} > 0$ then $\displaystyle\lim_{n \rightarrow \infty } S_{n} = \infty \quad\text{a.s.}$ using the strong law of large numbers, contradiction. But I can not show the case where $EX_{1} = 0$. Help?

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